document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1205063>Question 1205063</A>: <I><SPAN STYLE=\"word-break: break-all;\"> You were tasked to program an algorithm for a number generator. The generator will be used by a soda company for their bottlecaps. The soda company will be holding a bottlecap lottery where 5 select bottlecaps will be winning combinations. There would be three characters printed on the bottlecaps with each character having the chance to be repeated. If a second person was to join, how many bottles of Pepsi will the 2nd person need to buy to have a 1% chance at winning. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848161 by </B><A HREF=/tutors/aboutme.mpl?userid=textot><B>textot(100)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=textot><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848161\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Determine the Total Possible Combinations**\r<BR>\n" );
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document.write( "* Since there are 3 characters and each character can be repeated, the total number of possible combinations on a bottlecap is 3 * 3 * 3 = 27.\r<BR>\n" );
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document.write( "**2. Probability of Winning for the First Person**\r<BR>\n" );
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document.write( "* The first person needs to find 5 specific winning combinations out of 27 possible combinations. <BR>\n" );
document.write( "* The probability of winning for the first person on any given bottlecap is 5/27.\r<BR>\n" );
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document.write( "**3. Probability of Losing for the First Person**\r<BR>\n" );
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document.write( "* The probability of losing for the first person on any given bottlecap is 1 - (5/27) = 22/27.\r<BR>\n" );
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document.write( "**4. Probability of Losing Consecutively for the Second Person**\r<BR>\n" );
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document.write( "* Let 'n' be the number of bottles the second person needs to buy.<BR>\n" );
document.write( "* The probability of losing 'n' consecutive times is (22/27)^n.\r<BR>\n" );
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document.write( "**5. Determine the Number of Bottles for a 1% Chance of Winning**\r<BR>\n" );
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document.write( "* We want the probability of losing consecutively to be 99% (100% - 1%):<BR>\n" );
document.write( "    * (22/27)^n = 0.99\r<BR>\n" );
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document.write( "* Solve for 'n':<BR>\n" );
document.write( "    * n = log(0.99) / log(22/27) <BR>\n" );
document.write( "    * n ≈ 0.45 / -0.1823 <BR>\n" );
document.write( "    * n ≈ 2.47\r<BR>\n" );
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document.write( "* Since the second person can't buy a fraction of a bottle, they would need to buy **at least 3 bottles** to have a 1% chance of winning.\r<BR>\n" );
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document.write( "**Therefore, the second person needs to buy at least 3 bottles of soda to have a 1% chance of winning.**<BR>\n" );
document.write( " </SPAN>\n" );
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